2001 AIME I Problems/Problem 13
Problem
In a certain circle, the chord of a -degree arc is centimeters long, and the chord of a -degree arc is centimeters longer than the chord of a -degree arc, where The length of the chord of a -degree arc is centimeters, where and are positive integers. Find
Solution
Note that a cyclic quadrilateral in the form of an isosceles trapezoid can be formed from three chords of three -degree arcs and one chord of one -degree arc. The diagonals of this trapezoid turn out to be two chords of two -degree arcs. Let , , and be the chords of the -degree arcs, and let be the chord of the -degree arc. Also let be equal to the chord length of the -degree arc. Hence, the length of the chords, and , of the -degree arcs can be represented as , as given in the problem.
Using Ptolemy's theorem,
We can then apply the quadratic formula to find the positive root to this equation since polygons obviously cannot have sides of negative length.
x = \[\frac{-18 + \sqrt{18^2 + 4(84)}}{2}} (Error compiling LaTeX. Unknown error_msg)
x = \[\frac{-18 + \sqrt{660}}{2}} (Error compiling LaTeX. Unknown error_msg)
simplifies to which equals Thus, the answer is .
See also
2001 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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